Express a general solution to the given equation using Bessel functions of either the first or second kind. x²y" + xy' + (x² – 256) y = 0 Express any Bessel functions using the notation J,(x) and Y,u(x). y(x) =D
Express a general solution to the given equation using Bessel functions of either the first or second kind. x²y" + xy' + (x² – 256) y = 0 Express any Bessel functions using the notation J,(x) and Y,u(x). y(x) =D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Bessel Functions and General Solutions
This page is dedicated to explaining the use of Bessel functions to determine the general solution to a differential equation.
#### Problem Statement
You are given the following differential equation:
\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]
The task is to express a general solution for this equation using Bessel functions of either the first or second kind.
#### Solution Approach
1. **Identify the Standard Form:**
The given differential equation is of the standard form for Bessel's equation:
\[ x^2 y'' + x y' + (x^2 - \nu^2) y = 0 \]
Comparing this with the given equation:
\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]
We identify that \(\nu^2 = 256\), which gives \(\nu = 16\).
2. **Expression Using Bessel Functions:**
The general solution to the Bessel's differential equation is given by:
\[ y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x) \]
where \(J_\nu(x)\) and \(Y_\nu(x)\) are the Bessel functions of the first and second kinds, respectively, and \(c_1\) and \(c_2\) are arbitrary constants.
Therefore, for our specific case, where \(\nu = 16\),
\[ y(x) = c_1 J_{16}(x) + c_2 Y_{16}(x) \]
#### Final Solution
\[ y(x) = \boxed{ c_1 J_{16}(x) + c_2 Y_{16}(x) } \]
Here \( c_1 \) and \( c_2 \) are constants determined by initial conditions or boundary values specific to the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62cf0025-2a72-406a-bc17-39c2a0537bca%2Fa1c0973f-813e-41d0-8056-33cec4f99ed8%2Fcptosg_processed.png&w=3840&q=75)
Transcribed Image Text:### Bessel Functions and General Solutions
This page is dedicated to explaining the use of Bessel functions to determine the general solution to a differential equation.
#### Problem Statement
You are given the following differential equation:
\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]
The task is to express a general solution for this equation using Bessel functions of either the first or second kind.
#### Solution Approach
1. **Identify the Standard Form:**
The given differential equation is of the standard form for Bessel's equation:
\[ x^2 y'' + x y' + (x^2 - \nu^2) y = 0 \]
Comparing this with the given equation:
\[ x^2 y'' + x y' + \left( x^2 - 256 \right) y = 0 \]
We identify that \(\nu^2 = 256\), which gives \(\nu = 16\).
2. **Expression Using Bessel Functions:**
The general solution to the Bessel's differential equation is given by:
\[ y(x) = c_1 J_\nu(x) + c_2 Y_\nu(x) \]
where \(J_\nu(x)\) and \(Y_\nu(x)\) are the Bessel functions of the first and second kinds, respectively, and \(c_1\) and \(c_2\) are arbitrary constants.
Therefore, for our specific case, where \(\nu = 16\),
\[ y(x) = c_1 J_{16}(x) + c_2 Y_{16}(x) \]
#### Final Solution
\[ y(x) = \boxed{ c_1 J_{16}(x) + c_2 Y_{16}(x) } \]
Here \( c_1 \) and \( c_2 \) are constants determined by initial conditions or boundary values specific to the problem.
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